Homework 1 Solution
GIS in Water Resources
Fall 2012
Prepared by Gonzalo E. Espinoza
- Map Projection Parameters
(a) What earth datum is used in this coordinate system?
North American 1983
(b) What map projection is used in this coordinate system?
Lambert Conformal Conic
(c) Sketch on the map the standard parallels, the central meridian and the latitude of origin of this projection.
(d) For this projection, what are the coordinates of the origin (φo, λo) (in units of degrees minutes and seconds) and the corresponding (Xo, Yo) (in units of feet?
ϕ0,λ0=29°40'0"N;100°20'0"W
x0,y0=2,296,583.33;9,842,500.00ft
- Locations on the Earth
Austin ϕ=30°17'10"N=30+17/60+10/3600=30.2861°
λ=97°44'22"W=97+44/60+22/3600=-97.7394°
Lat,Long=30.2861°,-97.7394°
Logan ϕ=41°44'54"N=41+44/60+54/3600=41.7483°
λ=111°48'30"W=111+48/60+30/3600=-111.8083°
Lat,Long=41.7483°,-111.8083°
Lincoln ϕ=40°49'09"N=40+49/60+9/3600=40.8192°
λ=96°41'55"W=96+41/60+55/3600=-96.6986°
Lat,Long=40.8492°,-96.6986°
Distances on the Earth
a) Logan to Austin
Logan ϕA=41.7483°π180°=0.728646
λA=-111.8083°π180°=-1.951424
Austin ϕB=30.2861°π180°=0.528592
λB=-97.7394°π180°=-1.705875
R=6.371km
Dist=Rcos-1sinϕAsinϕB+cosϕAcosϕBcosλA-λB
Dist=1,791km
b) Austin to Lincoln
Austin ϕB=30.2861°π180°=0.528592
λB=-97.7394°π180°=-1.705875
Lincoln ϕB=40.8192°π180°=0.712429
λB=-96.6986°π180°=-1.687709
R=6.371km
Dist=Rcos-1sinϕAsinϕB+cosϕAcosϕBcosλA-λB
Dist=1,175km
c) Lincoln to Logan
Lincoln ϕB=40.8192°π180°=0.712429
λB=-96.6986°π180°=-1.687709
Logan ϕA=41.7483°π180°=0.728646
λA=-111.8083°π180°=-1.951424
R=6,371.0km
Dist=Rcos-1sinϕAsinϕB+cosϕAcosϕBcosλA-λB
Dist=1,265km
Slopes
a) Logan to Austin
S=4,789-6931,7910.30481,000=0.0007
The water would flow from Logan to Austin
b) Austin to Lincoln
S=693-1,1741,1750.30481,000=0.0001
The water would flow from Lincoln to Austin
c) Lincoln to Logan
S=1,174-4,7891,2650.30481,000=0.0009
The water would flow from Logan to Lincoln
- Web Mercator Coordinates
LatDD,LongDD=29.9942,-98.0886
The relationships between the Geographic Coordinates and the Web Mercator Coordinates are the following:
WebMercatorX=Semimajor Axis×LongDD
WebMercatorX=6,378,137m-98.0886π180
WebMercatorX=-10,919,173m≈-10,919,174m
WebMercatorY=Semiminor Axis×LongDD
WebMercatorY=6,356,752.31m29.9942π180
WebMercatorY=3,327,744m≠3,502,800m
The pair of coordinates in the X direction (Longitude) follows a simple relationship. In contrast, the pair of coordinates in the Y direction does not follow the simple relationship: Wm=Re×Angle. This means that for the projection, the X direction is conserved and the error is carried by the Y direction (with a correction). This is because the Mercator is a cylindrical map projection and in the cylinder is oriented from South to North.
- Sizes of DEM cells
Re=6,371.0km
∆λ=∆ϕ=18°=0.125°π180°=2.1817×10-3
a) Austin
ϕ=30.2861°=0.528592
AB=Re∆λcosϕ
AB=6,371.0km2.1817×10-3cos0.528592
AB=12.00km
AC=Re∆ϕ
AC=6,371.0km2.1817×10-3
AC=13.90km
ABCD=12.00×13.90=166.8km2
a) Logan
ϕ=41.7483°=0.728646
AB=Re∆λcosϕ
AB=6,371.0km2.1817×10-3cos0.728646
AB=10.37km
AC=Re∆ϕ
AC=6,371.0km2.1817×10-3
AC=13.90km
ABCD=10.37×13.90=144.1km2
a) Lincoln
ϕ=40.8192°=0.712429
AB=Re∆λcosϕ
AB=6,371.0km2.1817×10-3cos0.728646
AB=10.52km
AC=Re∆ϕ
AC=6,371.0km2.1817×10-3
AC=13.90km
ABCD=10.52×13.90=146.2km2